Properties

Label 439530.i
Number of curves $8$
Conductor $439530$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 439530.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
439530.i1 439530i7 \([1, 1, 0, -24439952583, 1470600247336053]\) \(13722604572968640187892492722921/36939806611960382108160\) \(4345931308090526994642915840\) \([2]\) \(955514880\) \(4.5376\) \(\Gamma_0(N)\)-optimal*
439530.i2 439530i8 \([1, 1, 0, -4343210183, -81360531892107]\) \(77013704252633562960444236521/20262661472595628847255040\) \(2383881859589403138250708200960\) \([2]\) \(955514880\) \(4.5376\)  
439530.i3 439530i5 \([1, 1, 0, -4019439008, -98085244614552]\) \(61042428203425827148268361721/2287149206968899000\) \(269080817050683998451000\) \([2]\) \(318504960\) \(3.9883\)  
439530.i4 439530i6 \([1, 1, 0, -1546525383, 22375778720373]\) \(3477015524751011858387583721/173605868128473455001600\) \(20424556779446773507483238400\) \([2, 2]\) \(477757440\) \(4.1910\) \(\Gamma_0(N)\)-optimal*
439530.i5 439530i4 \([1, 1, 0, -403729008, 537193271448]\) \(61859347930211625693801721/34737934177406743101000\) \(4086883218037725919089549000\) \([2]\) \(318504960\) \(3.9883\) \(\Gamma_0(N)\)-optimal*
439530.i6 439530i2 \([1, 1, 0, -251584008, -1527931671552]\) \(14968716721822395621081721/91209357028881000000\) \(10730689645090820769000000\) \([2, 2]\) \(159252480\) \(3.6417\) \(\Gamma_0(N)\)-optimal*
439530.i7 439530i1 \([1, 1, 0, -6584008, -51414671552]\) \(-268291321601301081721/9550359000000000000\) \(-1123590185991000000000000\) \([2]\) \(79626240\) \(3.2951\) \(\Gamma_0(N)\)-optimal*
439530.i8 439530i3 \([1, 1, 0, 59106617, 1368653011573]\) \(194108149567956675968279/6990401110687088640000\) \(-822413700271225291407360000\) \([2]\) \(238878720\) \(3.8444\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 439530.i1.

Rank

sage: E.rank()
 

The elliptic curves in class 439530.i have rank \(0\).

Complex multiplication

The elliptic curves in class 439530.i do not have complex multiplication.

Modular form 439530.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{12} - q^{13} + q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.